Creating a sinusoidal model using temperature v days of the year

•Topic: Sinusoidal functions

Aim: How can we use the graphing calculator to find the model of a sine function?

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•Write the equation of a sine function with a >0, amplitude 7.85, and cycles .016 in 2π  

•The sine function with shifts

•Y = A sin (Bx + C) + D

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•A = amplitude

•B = cycles in 2π

•C = horizontal shift

•D = vertical shift

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•Look at problem 27 on pg 368 handout

•Modeling temperature on several days of the year

•Graph the model

•Y1 = = 7.85 sin(.016x – 2.22) + 82.19

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•Fix the WINDOW for domain and range

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•GRAPH

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•Sketch the graph on the xy plane

•Extending your thinking

•How could we use the sine model to write the cosine model?

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•Recall how we shift from sine to cosine (by π/2)

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•Graph the cos model on the same set of axes

•Horizontal translations – phase shifts
g(x): horizontal translation of f(x)
g(x) = f (x – h)

•Important consideration

•The value of h f (x – h):

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•If h > 0, the shift is to the right

–Example: f ( x – 3 ) is a horizontal shift 3 units to the right

•If h < 0, the shift is to the left.

–Example: cos (x + 4) = cos (x – (-4)), h = -4; the shift is 4 units to the left

•Vertical translations
h(x): vertical translation of f(x)
h(x) = f(x) + k

•Exit slip

•What are the major differences between the cosine and the sine functions?

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translating sinusoidal functions

•Full period examination: Tuesday April 23

•Sine curve & Cosine curve

•Amplitude

•Period

•Range

•Sketching

•Solving sinusoidal equations

•Tides and temperature problems

•Writing the sinusoidal equation

•Topic: Sinusoidal functions

Aim: How can we graph the translations of sinusoidal functions?

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•Write an equation of a sine function with a >0, amplitude 2, and period π  

•Translating sine and cosine functions

Essential Understanding:

You can translate periodic functions in the same way that you translate other functions.

•You can graph f (x –h) by translating the graph of f by |h| units horizontally.

•You can graph f(x) + k by translating the graph of f by |k| units vertically.

•Horizontal translations – phase shifts
g(x): horizontal translation of f(x)
g(x) = f (x – h)

•Important consideration

•The value of h f (x – h):

•

•If h > 0, the shift is to the right

–Example: f ( x – 3 ) is a horizontal shift 3 units to the right

•If h < 0, the shift is to the left.

–Example: cos (x + 4) = cos (x – (-4)), h = -4; the shift is 4 units to the left

•Vertical translations
h(x): vertical translation of f(x)
h(x) = f(x) + k

•Got it?

•What is the value of h in each translation?

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•Describe each phase shift (use a phrase such as 3 units to the left).

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• a. g(t) = f(t – 5)

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• b. y = sin (x + 3)

•Graphing translations

A. y = sin x + 3

• k = 3

•Translate the graph of the parent function 3 units up

B.y = sin ( x – π/2)

  Translate the graph of the parent function π/2 units to the right

•Exit slip

•What are the major differences between the cosine and the sine functions?

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using the graphing calculator to analyze sinusoidal functions

•Full period examination: Tuesday April 23

•Sine curve & Cosine curve

•Amplitude

•Period

•Range

•Sketching

•Solving sinusoidal equations

•Tides and temperature problems

•Writing the sinusoidal equation

•Topic: Sinusoidal functions
Aim: Using the graphing calculator to analyze the sine and cosine equation

•Do Now:

•Solve the equation in the interval from 0 to 2π. Round your answer to the nearest hundredth:

•2 cos 3Θ = 1.5  

•Solving a cosine equation

Using the graphing calculator,

Y1 = 1.5

Y2 = 2 cos 3Θ  

•Graph both on the same screen

•Window function of the calculator

•Sets up the domain and range and scales units so that you can comfortably see and analyze the solutions of the equation:

•Xmin = 0

•Xmax = 2π

•Xscl = 1

•Ymin = -2

•Ymax = 2

•Yscl = 0.5

•Solving a sinusoidal equation

Use the INTERSECT feature to find the points at which the two graphs intersect.

The graphs show six solutions in the interval

(from 0 to 2π):

They are t ≈ 0.24, 1.85, 2.34, 3.95, 4.43, and 6.04.

These are the solutions of the equation based on the domain from 0 to 2π.

•Group work: Practice 13-5

•Complete Questions # 32, 33 first

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•Then choose any 3 of the circles problems to complete.

•Each question is worth 20 points.

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•This will be counted as an assessment.

•Exit slip

•What are the major differences between the cosine and the sine functions?

•Answer to the DNA question

•Because one cos curve is drawn from 0 to 24 (x-axis), the period of the curve is 24.

•Using the formula y = a cos bx, we can find a = 1 (amplitude) and b = π/12 (because period = 2π/b, so 24 = 2π/b, making b = π/12.)

•Therefore, y = 1cos (π/12)x or simply y = cos (π/12)x .

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solving both cosine and sine equations using the graphing calculator

•Topic: Sinusoidal functions

Aim: How can we apply what we know about solving cosine equations to solving sine equations? 

•Do Now

•Identify the period, range and amplitude of y = 3 cos Θ   

•Think about a plan

•In Buenos Aires, Argentina, the average monthly temperature is highest in January and lowest in July, ranging from 83° F to 57° F.

•Write a cosine function that models the change in temperature according to the month of the year.

•Step 1: Find amplitude = ½ (83 – 57) = 13

•Step 2: find the period = 12 months

•Step 3: Average temperature = ½ (140) = 70°F

•

•Write the Cosine function

•Y = a cos b t

•Where t = # of months since January

•January is where t = 0

•Y = 13 cos b t

•Period = 12

•12 = 2π/b

•b = π/6 (can you explain why?)

•Cosine equation is y = 13 cos π/6 t

•Lets graph the cosine equation over a two year period

•Y = 13 cos π/6 t

•Use the graphing calculator

•Window:

•Xmin = 0

•Xmax = 24

•Xscl = 2

•Ymin = -13

•Ymax = 13

•Yscl = 2

•Y1 = 13 cos πt/6 then press “graph”

•

•

•What is the temperature in August of the first year?

•Use equation y = 13 cos πt / 6

•Where t = 7 (why is t = 7?)

•Solve y = 13 cos 7π/6

•Method 1: Use graphing calculator

2nd calc, value, x = 7, enter

Y = -11.25833

Method 2: type in 13 cos (7π/6) = -11.25833

Solution --- temp in August is 58.75° F.

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•Analysis of temperature in August

•Recall that the average temp is  70°F

•Since y = -11.25 which is the change in temp from the average, the temp in august is 70 – 11.25 = 58.75°F.

•In class assessment

•Problem #36: solving a sine equation using a cosine equation

•Complete problems 10, 11, 18, 19, 21, 27 and 30 on the handout.

•Page 857 circled problems

•All questions should be answered on separate sheet of paper with your name clearly displayed

•Hand in before the bell rings

•Exit slip

•What are the solutions to the following equation in the interval from 0 to 2π?

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•-2 cos Θ = -1.2

•Answer to the DNA question

•Because one cos curve is drawn from 0 to 24 (x-axis), the period of the curve is 24.

•Using the formula y = a cos bx, we can find a = 1 (amplitude) and b = π/12 (because period = 2π/b, so 24 = 2π/b, making b = π/12.)

•Therefore, y = 1cos (π/12)x or simply y = cos (π/12)x .

•

cosine function to model changes in the tides

•Topic: Sinusoidal functions
Aim: How can we create a cosine equation modeling a “tide” problem?

•Do Now:

•If y = -2 cos Θ

•What is the amplitude and period of the function?  

•Sketching a cosine graph

•Identify amplitude

•Identify the cycle, b

•Use period = 2π/b

•Draw the xy plane and label your axes

•Decide on the units for each axis and label

•Identify and plot the 5 point summary

•Connect the points and voila !!!

•Tides

•Average depth = 5.5 ft

•Amplitude = 1.5 ft

•12 hr and 22 min for one cycle

•Y = 1.5 cos 60π/371 t

•Suppose your boat needs at least 5 ft of water to approach or leave the  pier, what equation would model this situation?

Can you find these times?

•In class assessment

•Complete problems 10, 11, 18, 19, 21, 27 and 30 on the handout.

•Page 857 circled problems

•All questions should be answered on separate sheet of paper with your name clearly displayed

•Hand in before the bell rings

•Exit slip

•What are the solutions to the following equation in the interval from 0 to 2π?

•-2 cos Θ = -1.2

•Answer to the DNA question

•Because one cos curve is drawn from 0 to 24 (x-axis), the period of the curve is 24.

•Using the formula y = a cos bx, we can find a = 1 (amplitude) and b = π/12 (because period = 2π/b, so 24 = 2π/b, making b = π/12.)

•Therefore, y = 1cos (π/12)x or simply y = cos (π/12)x .

Sinusoidal phase shift - 04/09//13

Topic: Sinusoidal functions

Aim: How can we represent a horizontal shift of the sine function?

Do Now: Sketch a sine curve with an amplitude of 1 and a period of 2π.

Analyzing the basic sine curve

In your sketch:

a) identify the maximum value

b) identify the minimum value

c) identify the ‘three’ zeros

　

Modeling a horizontal shift

y = sin x              y = cos x

Horizontal shift – phase shift

The graphs of sine and cosine are the same when sine is shifted left by

π/2. Such a shifting is referred to as a horizontal shift (or phase shift).

Shift sine to the left to create cosine.

(Shift cosine to the right to create sine)

On your own

Start with your sketch of y = sin x

Shift to the left by

π/2 and sketch a basic cos curve (use a different color)

Your sketch should resemble diagram in previous slide.

Analyzing the basic cos curve

In your sketch:

a) identify the maximum value

b) identify the minimum value

c) identify the zeros (how many are there?)

Equivalent sinusoidal function

　

Analysis

If the horizontal shift is positive, the shifting moves to the right.  If the horizontal shift is negative, the shifting moves to the left.  

y = A sin( B (x -C) ) + D

From the sinusoidal equation above, the horizontal shift is obtained by determining the change being made to the x value. The horizontal shift is C.

Note: when C is negative, the shift is right and when C is positive, the shift is left

Exit slip

If you graph the basic sine function, y = sin x and then graph y = sin (x +

π/2), how will it transform?

What kind of shift has occurred?

Can you predict what a vertical shift might look like?